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Theorem cfinufil 22060
Description: An ultrafilter is free iff it contains the Fréchet filter cfinfil 22025 as a subset. (Contributed by NM, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
cfinufil (𝐹 ∈ (UFil‘𝑋) → ( 𝐹 = ∅ ↔ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ Fin} ⊆ 𝐹))
Distinct variable groups:   𝑥,𝐹   𝑥,𝑋

Proof of Theorem cfinufil
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elpwi 4359 . . . . 5 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
2 ufilb 22038 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (¬ 𝑥𝐹 ↔ (𝑋𝑥) ∈ 𝐹))
32adantr 473 . . . . . . . . 9 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) ∧ (𝑋𝑥) ∈ Fin) → (¬ 𝑥𝐹 ↔ (𝑋𝑥) ∈ 𝐹))
4 ufilfil 22036 . . . . . . . . . . . 12 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
54adantr 473 . . . . . . . . . . 11 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → 𝐹 ∈ (Fil‘𝑋))
6 filfinnfr 22009 . . . . . . . . . . . . 13 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑋𝑥) ∈ 𝐹 ∧ (𝑋𝑥) ∈ Fin) → 𝐹 ≠ ∅)
763exp 1149 . . . . . . . . . . . 12 (𝐹 ∈ (Fil‘𝑋) → ((𝑋𝑥) ∈ 𝐹 → ((𝑋𝑥) ∈ Fin → 𝐹 ≠ ∅)))
87com23 86 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘𝑋) → ((𝑋𝑥) ∈ Fin → ((𝑋𝑥) ∈ 𝐹 𝐹 ≠ ∅)))
95, 8syl 17 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ Fin → ((𝑋𝑥) ∈ 𝐹 𝐹 ≠ ∅)))
109imp 396 . . . . . . . . 9 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) ∧ (𝑋𝑥) ∈ Fin) → ((𝑋𝑥) ∈ 𝐹 𝐹 ≠ ∅))
113, 10sylbid 232 . . . . . . . 8 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) ∧ (𝑋𝑥) ∈ Fin) → (¬ 𝑥𝐹 𝐹 ≠ ∅))
1211necon4bd 2991 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) ∧ (𝑋𝑥) ∈ Fin) → ( 𝐹 = ∅ → 𝑥𝐹))
1312ex 402 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ Fin → ( 𝐹 = ∅ → 𝑥𝐹)))
1413com23 86 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ( 𝐹 = ∅ → ((𝑋𝑥) ∈ Fin → 𝑥𝐹)))
151, 14sylan2 587 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → ( 𝐹 = ∅ → ((𝑋𝑥) ∈ Fin → 𝑥𝐹)))
1615ralrimdva 3150 . . 3 (𝐹 ∈ (UFil‘𝑋) → ( 𝐹 = ∅ → ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹)))
174adantr 473 . . . . . . . . . . . 12 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → 𝐹 ∈ (Fil‘𝑋))
18 uffixsn 22057 . . . . . . . . . . . 12 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → {𝑦} ∈ 𝐹)
19 filelss 21984 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ {𝑦} ∈ 𝐹) → {𝑦} ⊆ 𝑋)
2017, 18, 19syl2anc 580 . . . . . . . . . . 11 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → {𝑦} ⊆ 𝑋)
21 dfss4 4059 . . . . . . . . . . 11 ({𝑦} ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ {𝑦})) = {𝑦})
2220, 21sylib 210 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (𝑋 ∖ (𝑋 ∖ {𝑦})) = {𝑦})
23 snfi 8280 . . . . . . . . . 10 {𝑦} ∈ Fin
2422, 23syl6eqel 2886 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin)
25 difss 3935 . . . . . . . . . . 11 (𝑋 ∖ {𝑦}) ⊆ 𝑋
26 filtop 21987 . . . . . . . . . . . 12 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
27 elpw2g 5019 . . . . . . . . . . . 12 (𝑋𝐹 → ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ {𝑦}) ⊆ 𝑋))
2817, 26, 273syl 18 . . . . . . . . . . 11 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ {𝑦}) ⊆ 𝑋))
2925, 28mpbiri 250 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋)
30 difeq2 3920 . . . . . . . . . . . . 13 (𝑥 = (𝑋 ∖ {𝑦}) → (𝑋𝑥) = (𝑋 ∖ (𝑋 ∖ {𝑦})))
3130eleq1d 2863 . . . . . . . . . . . 12 (𝑥 = (𝑋 ∖ {𝑦}) → ((𝑋𝑥) ∈ Fin ↔ (𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin))
32 eleq1 2866 . . . . . . . . . . . 12 (𝑥 = (𝑋 ∖ {𝑦}) → (𝑥𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
3331, 32imbi12d 336 . . . . . . . . . . 11 (𝑥 = (𝑋 ∖ {𝑦}) → (((𝑋𝑥) ∈ Fin → 𝑥𝐹) ↔ ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
3433rspcv 3493 . . . . . . . . . 10 ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 → (∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹) → ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
3529, 34syl 17 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹) → ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
3624, 35mpid 44 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹) → (𝑋 ∖ {𝑦}) ∈ 𝐹))
37 ufilb 22038 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ {𝑦} ⊆ 𝑋) → (¬ {𝑦} ∈ 𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
3820, 37syldan 586 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (¬ {𝑦} ∈ 𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
3918pm2.24d 149 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (¬ {𝑦} ∈ 𝐹 → ¬ 𝑦 𝐹))
4038, 39sylbird 252 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → ((𝑋 ∖ {𝑦}) ∈ 𝐹 → ¬ 𝑦 𝐹))
4136, 40syld 47 . . . . . . 7 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹) → ¬ 𝑦 𝐹))
4241impancom 444 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹)) → (𝑦 𝐹 → ¬ 𝑦 𝐹))
4342pm2.01d 182 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹)) → ¬ 𝑦 𝐹)
4443eq0rdv 4175 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹)) → 𝐹 = ∅)
4544ex 402 . . 3 (𝐹 ∈ (UFil‘𝑋) → (∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹) → 𝐹 = ∅))
4616, 45impbid 204 . 2 (𝐹 ∈ (UFil‘𝑋) → ( 𝐹 = ∅ ↔ ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹)))
47 rabss 3875 . 2 ({𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ Fin} ⊆ 𝐹 ↔ ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹))
4846, 47syl6bbr 281 1 (𝐹 ∈ (UFil‘𝑋) → ( 𝐹 = ∅ ↔ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ Fin} ⊆ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wne 2971  wral 3089  {crab 3093  cdif 3766  wss 3769  c0 4115  𝒫 cpw 4349  {csn 4368   cint 4667  cfv 6101  Fincfn 8195  Filcfil 21977  UFilcufil 22031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1o 7799  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-fbas 20065  df-fg 20066  df-fil 21978  df-ufil 22033
This theorem is referenced by: (None)
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